It is known that the appropriate reparameterizations for the parameter \(t\)  is necessary for a rigorous discussion in the space of probability measures [1]}, [2]}.
However, as mentioned in the literature [1]}, an explicit expression for the reparametrizations \(\tau _{p,a}\)  and \(\tau _{p,q}\)  is unknown.
A similar discussion has been made in the derivation of the \(\phi _{\beta }\) -path [4]}, where it is mentioned that the normalizing factor is unknown in general.
Furthermore, the \(f\) -mean is not convex depending on the \(\alpha \) .
For these reasons, it is generally difficult to discuss \(\alpha \) -geodesics in probability measures by normalization or reparameterization, and to avoid unnecessary complexity, the parameter \(t\)  is assumed to be appropriately reparameterized.
